How can I render animations in PS1 / 3D pixel style?
It's several things, color sets, modelisation styles, ... But on the technical side, it's basically about low resolution rendering.

Assuming you're using Blender Cycles :
First, you have to make sure that you use only low res textures on your model, with their pixel filtering set to "closest" (otherwise the default behavior is to "hide" the pixels by blur) :



Then, you shall render with low resolution, and make sure your render pixel filtering (which serves as a kind of anti-aliasing in raytracer renderers like cycles) is set to the lowest value in the render options :